New Trends in Differential Equations, Control Theory and Optimization

The following sections are included:

• Preface
• Organizing Committee of the Congress
• Contents

We survey some existence and multiplicity results for Dirichlet problems with mean curvature operator in Minkowski space. We deal with positive radial solutions when the problem has radial symmetry, as well as nontrivial constant sign solutions for problems in general bounded domains. Discontinuous Dirichet problems are also considered. Our approaches combine variational techniques, lower and upper solutions method and topological degree arguments.

We consider a PDE-constrained optimization problem governed by a free boundary fluid-elasticity interaction. The challenge of applying optimization tools to free boundary fluid-structure interactions is the proper derivation of the adjoint sensitivity information with correct balancing conditions on the common interface. As the interaction is a coupling of Eulerian (fluid state) and Lagrangian (motion of the solid) quantities, sensitivity analysis on the system falls into the framework of shape analysis. This paper builds upon the sensitivity theory developed in^{5, 6, 8} and provides a description for the adjoint linearized model, necessary in the derivation of optimality conditions for controls and derivative-based optimization algorithms in free boundary fluid-structure interactions.

In this note we review some results recently obtained for a model which describes diffusion of species using a suitable regularization of a “forwardbackward” parabolic equation. Existence and uniqueness of solutions, as well as continuous dependence on data, are derived for a system of partial differential equations, which renders the evolution of physical quantities as concentration and chemical potential.

We investigate second order dynamical systems of the from , where Γ : ℋ → ℋ is an elliptic bounded self-adjoint linear operator defined on a real Hilbert space ℋ, B : ℋ→ℋ is a cocoercive operator and λ: [0,+∞) → (0,+∞) is a relaxation function depending on time. We prove via Lyapunov analysis that the generated trajectories weakly converge to a zero of the operator B. This opens the gate towards approaching through a second order dynamical system the problem of determining zeros of the sum of a maximally monotone operator and a cocoercive one, which captures as particular case the minimization of the sum of a nonsmooth convex function with a smooth convex one. Finally, when B is the gradient of a smooth convex function, we prove a rate of 𝒪(1/t) for the convergence of the function values along the ergodic trajectory to its minimum value.

The existence of solutions of a quadratic integral inclusion is obtained in the case when the inclusion is defined by a Carathéodory set-valued map with convex values.

In this work is considered a system of coupled matrix nonlinear differential equations arising in connection with the solution of a zero sum two players linear quadratic differential game for a system modeled by an Itô differential equation subject to random switching according with a standard homogeneous Markov process with a finite number of state. The system of differential equations under consideration contains as special cases the game theoretic Riccati differential equations arising in the solution of the H_∞ control problem from the deterministic and stochastic cases. A set of sufficient conditions which guarantee the existence of the bounded and stabilizing solution of this kind of Riccati differential equations is provided.

In this paper we derive a maximum-type principle for a certain class of differential operators from some earlier results related to an eigenvalue problem involving the p(·)-Laplacian.

The main results of this paper concern integrability and differentiability properties for solutions for a nonlocal system. The boundary value problem is associated with an integro-differential equation that exhibits a weakly singular kernel. The main result provides a nonlocal equivalent of classical regularity theorems established for elliptic systems. In addition, we show that solutions inherit additional integrability properties of the forcing term, thus providing a simple proof to well-posedness of solutions in L^p for p > 2.

In this note we summarize some results obtained in¹⁹ and in²⁰, where starting from a non-convex and nonlocal 3D-variational model for the electric polarization in a ferroelectric material, via an asymptotic process we obtain 2D-variational models for a thin film and 1D-variational models for a thin wire, respectively.

The present contribution regards a sliding mode control problem for a phasefield model of Caginalp type and summarizes the results of a joint work with Viorel Barbu, Pierluigi Colli, Gabriela Marinoschi and Elisabetta Rocca obtained in the framework of the three year Italian-Romanian project on Nonlinear PDEs with applications in modeling cell growth, chemotaxis and phase transition.

Viable sets in multi-valued dynamics, generated by finite families of mappings, are considered. In the case of affine Iterated Function Systems, satisfying a generalized cone condition, the existence of compact viable sets is stated, and some topological characteristics of these sets are given.

The aim of this short note is to give a linear programming approach to optimality conditions (expressed through support criteria) in control problems with piecewise deterministic Markov dynamics. Two classes are considered: classical, discounted control problems and asymptotic problems associated to two-scales (perturbed) systems.

We are concerned with some inverse problems for abstract differential equations. Their motivation comes from relevant equations of interest in the control theory. Various concrete examples are described.

We consider active scalar equations with constitutive laws that are odd and very singular, in the sense that the velocity field loses more than one derivative with respect to the active scalar. We provide an example of such a constitutive law for which the equation is ill-posed: Either Sobolev solutions do not exist, from Gevrey-class datum, or the solution map fails to be Lipschitz continuous in the topology of a Sobolev space, with respect to Gevrey class perturbations in the initial datum.

This paper analyzes individual optimal vaccination strategies for a Susceptible-Infected-Recovered (SIR) model. The individuals vaccinate according to a criterion taking into account the risk of infection, the possible side effects of the vaccine and the overall epidemic course. We treat the situation when the vaccination capacity is limited and the individual discounts the future at some rate 𝒟 > 0. Since the course of the epidemic is determined by the sum of individual decisions a natural question is whether an equilibrium between the individual decisions and the epidemic evolution exists. We show that the answer is positive.

For a large class of constrained minimization problems in R^N invariant under translations, we prove that dichotomy of minimizing sequences may occur only if vanishing also occurs. Then we discuss the dependence of the Lagrange multipliers associated to minimizers with respect to the constraint and we show that continuity cannot be expected. Finally, we show that if we minimize under Pohozaev constraints we may control the Lagrange multipliers.

We survey some recent results on the existence of at least N +1 geometrically distinct periodic solutions for periodically forced N-dimensional relativistic pendulum differential and difference systems.

Our aim in this paper is to study the well-posedness and the dissipativity of the anisotropic Caginalp phase-field system with singular nonlinear terms.

The article contains an overview of our recent results about initial value problems with fractal filling space attractors and about fully discrete models for nonlinear anomalous diffusions on synchronized space-time grids.

In a real Hilbert space H we study the behavior of solutions u_εδ to the following Cauchy problem

The aim of this short note to give a summary of some recent results on global controllability of the viscous Burgers equation and mixing properties of the associated stochastic problem. We also outline the main ideas of the proofs. The details are given in^1,2.

We discuss the case of shape optimization problems with boundary cost functional. We investigate theoretical properties and an important tool is the implicit parametrization theorem.

We present some recent results on the existence of two-dimensional travelling free surface gravity water waves of large amplitude that may have critical layers or overhanging profiles.

The following sections is included:

• Author Index